The  Choice  of  the  Most  Probable  Value  for 
an  Atomic  Weight:  The  Atomic 
Weight  of  Hydrogen. 


By  William  A*  Noyes. 


5<K£. 


NV\ 


[Reprinted  from  the  Journal  of  the  American  Chemical  Society, 
Vol.  XXX,  No.  i,  January,  1908.] 


THE  CHOICE  OF  THE  MOST  PROBABLE  VALUE  FOR  AN  ATOMIC 
WEIGHT:  THE  ATOMIC  WEIGHT  OF  HYDROGEN.1 


By  William  A.  Noyes. 
Received  December  6,  1907. 


A large  amount  of  material  has  been  accumulated  from  which  the 
atomic  wieghts  of  the  more  important  elements  can  be  calculated.  A 
very  superficial  examination  of  this  material  reveals  the  fact  that  the 
experimental  results  on  which  our  knowledge  of  these  constants  is  based, 
vary  very  greatly  in  their  value  and  that  many  of  the  older  determina- 
tions have  been  rendered  practically  worthless  by  recent  work,  which 
has  been  more  careful  and  accurate. 

As  some  of  these  new  determinations  affect  the  values  for  elements 
of  such  fundamental  importance  that  a recalculation  of  the  whole  table 
of  atomic  weights  will  be  necessary  in  the  near  future,  it  seems  desirable 
to  formulate  some  general  principles  to  aid  in  the  elimination  of  results 
which  have  little  or  no  value  and  in  the  combination  of  the  results  which 
remain.  Such  principles,  if  they  meet  with  general  acceptance,  will 
be  of  value,  riot  only  for  the  purpose  stated  but  also  as  setting  a certain 
standard  which  must  be  attained  by  future  workers  in  this  field,  if  their 
work  is  to  be  of  permanent  value. 

The  most  important  general  principle  which  has  been  proposed  for 
the  combination  of  the  results  of  different  observers,  is  the  one  based  on 
the  mathematical  discussion  of  accidental  errors  of  observation.  In  ac- 
cordance with  the  theory  of  probabilities,  these  results,  if  subject  only 
to  accidental  errors,  should  be  weighted  in  inverse  proportion  to  their 

1 Presented  in  abstract  at  the  N.  Y.  Meeting  of  the  American  Chemical  Society, 
Dec.  28,  1906. 


the  atomic  weight  of  hydrogen. 


5 


probable  errors.1  A very  serious  objection  to  this  method  of  treatment 
lies  in  the  fact  that  every  determination  of  this  kind  is  subject  to  con- 
stant errors,  and  that  the  amount  of  these  errors  is  not  proportional  to 
their  “probable  errors.”2  Thus  Stas  obtained  132.8445  ±0.0008  parts 
of  silver  chloride  from  100  parts  of  silver,  while  Richards  and  Wells3 
have  obtained  132.8670  ±0.0005  parts.  The  most  probable  value 
calculated  by  the  mathematical  rule  would  be  132.8607.  If  this  value 
is  the  true  one,  the  real  error  of  the  value  obtained  by  Richards  and 
Wells  is  12  times  its  probable  error,  while  the  real  error  of  Stas  is  20  times 
the  probable  error.  And,  whatever  the  true  value  may  be,  the  real  error 
of  one  of  the  results,  at  least,  is  many  times  its  “probable  error.”  An 
examination  of  other  cases  shows  that  the  relations  here  found  are  typi- 
cal, and  it  seems  evident  that  the  question  of  constant  errors  requires 
some  other  treatment  than  the  simple  mathematical  one.  The  proper 
treatment,  which  is  an  experimental  one,  has  been  clearly  illustrated 
in  the  case  which  we  have  under  consideration.  Richards  and  Wells 
studied  their  method  very  carefully  with  especial  reference  to  the  elimina- 
tion of  constant  errors  and  to  secure  evidence  as  to  the  amount  of  those 
errors  which  could  not  be  wholly  excluded.  They  also  pointed  out 
certain  errors  in  the  work  of  Stas  and  determined,  approximately  the 
magnitude  of  some  of  these.  It  is  evident  for  this  reason  that  very  much 
greater  weight  attaches  to  the  value  found  by  Richards  and  Wells  than 
to  that  found  by  Stas,  and  it  is  proposed  as  a general  principle  that  when 
a later  observer  has  pointed  out  sources  of  error  which  are  considerable 
in  comparison  with  the  “probable  errors”  and  where  the  later  observer 
has  succeeded  in  avoiding  these  sources  of  error,  the  earlier  work  must 
be  looked  upon  as  having  only  confirmatory  value  and  the  result  of  the 
later  work  should  be  accepted  without  modification.  It  has  been  ob- 
jected to  this  that  the  later  work  is  also  subject  to  constant  errors  which 
may  be  in  the  opposite  direction  from  those  of  the  earlier  determina- 
tion and  that  if  we  give  a certain  weight  to  the  earlier  work  we  may 
eliminate  these  errors.  But  we  certainly  are  not  justified  in  using  a value 
that  contains  a known  error  in  one  direction  merely  for  the  chance  that 
we  may  compensate  an  unknown  error. 


of  the  atomic  weights  are  nearer  to  the  numbers  nov 
than  were  the  values  then  given  by  Professor  Clarke. 

^ TV»1  C Tmirtiol  h*  r-  . . 


This  Journal,  27,  524. 


(,.19  2.9  5 


6 


WILLIAM  a.  NOYES. 


The  principle  outlined  above  has  been  recently  proposed,  independ- 
ently, by  Professar  Guye,1  in  his  discussion  of  the  selection  of  the  most 
probable  value  for  the  density  of  a gas.  A second  principle  proposed  by 
Professor  Guye  is  that  when  the  values  obtained  by  two  observers  agree 
while  that  obtained  by  a third  observer  is  discordant,  the  values  which 
are  in  agreement  should  be  given  much  greater  weight.  As  an  exten- 
sion of  this  principle,  a value  of  an  individual  worker  which  differs  ma- 
terially from  the  values  obtained  by  several  others,  should  be  rejected 
entirely. 

After  eliminating  the  results  which  are  excluded  by  the  application 
of  the  foregoing  principles,  it  is  proposed  to  arrange  those  which  remain 
in  the  order  of  their  probable  errors.  Any  result  with  a probable  error 
more  than  five  times  that  of  the  smallest  probable  error  may  be  excluded, 
as  such  a result  will  have  only  one  twenty- fifth  the  weight,  according 
to  the  theory  of  probabilities.  In  practical  effect,  this  is  the  same  as  using 
the  mathematical  rule  which  Professor  Clarke  has  so  long  employed 
in  weighting  the  results  of  different  workers.  As  at  least  five  or  six  ob- 
servations are  necessary  to  give  a probable  error  which  has  any  sig- 
nificance, results  based  on  a smaller  number  of  determinations  may  be 
excluded  unless  other  evidence  warrants  the  belief  that  the  work  is  of  an 
unusual  degree  of  accuracy. 

The  values  for  any  given  ratio  which  remain  after  the  elimination 
of  results  which  have  little  value,  may  well  be  combined  by  weighting 
them  inversely  as  the  squares  of  their  probable  errors. 

For  further  use,  the  ratios  which  are  selected  in  this  manner  should  be 
weighted,  not  by  the  probable  error  calculated  by  the  mathematical 
rule  but  by  the  deviation  of  the  results  of  different  observers  from  the 
value  selected.  If  the  results  of  only  one  observer  remain  after  elimina- 
ting untrustworthy  values  (as  in  the  case  of  the  ratio  of  silver  to  silver 
chloride),  this  result  should  be  weighted  in  accordance  with  the  average 
deviation  of  the  results  of  this  observer  from  his  mean.  This  will,  I 
think,  give  a much  fairer  basis  than  the  “probable  error”  for  weighting 
the  value  in  such  cases.  Thus  the  “mean  error”  of  the  value  of  Rich- 
ards and  Wells  given  above  is  0.0018,  while  the  “ probable  error”  is  0.0005. 
When  we  consider  the  certainty  that  some  sources  of  constant  error 
will  always  remain,  I think  every  one  will  agree  that  the  real  error  is 
much  more  likely  to  correspond  to  the  former  than  to  the  latter  value. 

After  selecting  the  most  trustworthy  experimental  ratios  as  suggested, 
we  have  still  to  combine  them  for  the  calculation  of  atomic  weights. 
This  may  usually  be  done  in  a variety  of  ways.  In  choosing  among 
these,  the  same  general  principles  as  before  should  be  applied.  For 
a given  atomic  weight,  only  those  ratios  should  be  used  for  which  the  un- 
1 Arch.  sci.  phys.  nat.,  24,  44. 


THE  ATOMIC  WEIGHT  OF  HYDROGEN. 


n 


certainties  of  the  values  will  affect  the  atomic  weight  chosen  less  than 
five  times  as  much  as  any  other  combination  of  ratios  which  might  be 
used. ' In  most  cases  this  will  probably  lead  to  the  selection  of  ratios 
which  furnish  a direct  comparison  with  oxygen,  silver  or  one  of  the  halo- 
gens rather  than  of  those  in  which  the  comparison  is  more  indirect.  Den- 
sities of  gases  corrected  to  the  condition  of  an  ideal  gas  by  the  method 
of  D.  Berthelot1  may  be  considered  as  direct  comparisons  with  oxygen, 
and  molecular  and  atomic  weights  calculated  from  these  densities  should 
be  included  with  those  determined  by  chemical  methods. 

The  Atomic  Weight  of  Hydrogen. 

The  following  is  a summary  of  the  determinations  which  have  been  made 
of  the  atomic  weight  of  hydrogen  by  the  chemical  method : 


No.  of 

Prob. 

Real 

Real  error 

Date. 

expts. 

Value. 

error. 

error. 

Prob.  error' 

Dulong  and  Berzelius 

1821 

3 

I .00667 

356 

108 

0-3 

Leduc 

1892 

2 

I .00749 

83 

26 

0-3 

Erdmann  and  Marchand 

1842 

8 

I .00160 

71 

615 

8.7 

Thomsen 

1870 

8 

I.00570 

7 1 

205 

2.9 

Rayleigh 

1889 

5 

I .00692 

56 

85 

i-5 

Dumas 

1842 

19 

1 00250 

44 

525 

12  .0 

Keiser 

1898 

8 

I.OO753 

3i 

22 

0.7 

Dittmar  and  Henderson.  . . . 

1890 

24 

I . 00840 

29 

65 

2 . 2 

Noyes  (recalculated) 

1890 

24 

1 .00765 

17 

IO 

0.6 

Thomsen 

1895 

21 

I .00826 

14 

51 

3-6 

Cooke  and  Richards 

1887 

16 

I .00826 

13 

5i 

3-9 

Noyes  (original) 

1890 

24 

I .00654 

1 1 

121 

1 1 .0 

Keiser 

1888 

10 

I . 00306 

7 

469 

67.0 

Noyes 

1907 

48 

I .00787 

2 

12 

6.0 

Morley 

1895 

23 

I .00762 

2 

12 

6.0 

The  probable  errors  of  the 

table 

are 

calculated 

from  those  assigned 

by  Professor  Clarke.2  For  the  results  of  Erdmann  and  Marchand  and 
Leduc,  the  values  are  arbitrary.  For  convenience  these  errors  are  given 
in  units  corresponding  to  the  last  significant  figure  of  the  values  for  the 
atomic  weights. 

On  applying  the  principles  which  have  been  outlined,  we  find  that 
the  results  of  Dulong  and  Berzelius,  Erdmann  and  Marchand,  and  of 
Dumas,  are  excluded  because  the  later  work  of  Dittmar  and  Hender- 
son by  the  same  method,  demonstrates  that  serious  constant  errors  were 
involved  in  the  earlier  work.  Leduc’s  value  is  to  be  rejected  because 
the  number  of  experiments  was  too  small.  Reiser’ s earlier  value  is  to 
be  rejected  because  it  is  not  in  accord  with  any  of  the  later  work  and  be- 
cause he  has  himself  given  us  a later  and  better  value.  My  own  original 
value  must  be  rejected  because  it  was  subject  to  a constant  error  and 

1 Compt.  rend.,  144,  76. 

2 “Constants  of  Nature.”  Part  V.,  p.  24  (1897). 


8 


WILLIAM  a.  NOYES. 


the  recalculated  result  may  be  considered  as  superseded  by  my  later 
and  more  careful  work.  Because  the  probable  errors  of  all  of  the  other 
determinations  are  more  than  five  times  as  great  as  those  of  Morley  and 
myself,  they  would  be  excluded  by  the  third  principle  proposed.  The 
final  value,  if  calculated  from  these  two  results,  is  1.00775, 

It  is  interesting  to  notice  the  relation  between  the  real  errors  of  the 
various  values  (assuming  this  value  as  true)  and  the  probable  errors. 
Only  in  those  cases  where  we  now  know  that  there  were  serious  constant  er- 
rors, is  the  real  error  more  than  six  times  the  probable  error. 

Morley  calculates  a value  corresponding  to  1.00762  from  his  deter- 
minations of  the  densities  of  the  gases  and  their  combining  volumes. 
This  value  has  not  been  considered  here,  partly  because  the  probable 
error  of  the  density  of  hydrogen  is  about  3 in  100,000,  instead  of  2 for 
the  chemical  method,  but  chiefly  because  of  the  uncertainty  of  the  ratio 
of  the  combining  volumes. 1 

If  a value  is  calculated  by  Professor  Clarke’s  method,  weighting  each 
result  in  inverse  proportion  to  its  probable  error,  only  Keiser’s  older 
value  and  my  own  original  value  would  affect  the  value  which  I have  se- 
lected by  more  than  about  one  part  in  100,000.  Keiser’s  older  value  would, 
however,  reduce  it  by  about  40  parts  and  my  own  original  value  by  about 
4 parts  in  100,000. 

1 Morley:  “Smithsonian  Contribution  to  Knowledge,”  No.  980,  p.  no  (1895). 

University  of  Illinois. 

Urbana,  III. 


